In this talk, we will characterize Chern classes of vector bundles on schemes and discuss and invoke the splitting principle as a tool for computing Chern class identities for tensor/wedge/symmetric products of bundles. Time permitting, we may either return to Grassmannians to do some computations or say some words about how Chern classes of the universal bundle generate the Chow ring, or talk about the Chow ring of a projective bundle and its relation to the Chern classes of the underlying vector bundle.